Detection-device-independent verification of nonclassical light
DDetection-device-independent verification of nonclassical light
Martin Bohmann, ∗ Luo Qi,
2, 3
Werner Vogel, and Maria Chekhova
2, 3, † INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy Max-Planck-Institute for the Science of Light, Erlangen, Germany University of Erlangen-N¨urnberg, Staudtstrasse 7/B2, D-91058 Erlangen, Germany Arbeitsgruppe Theoretische Quantenoptik, Institut f¨ur Physik, Universit¨at Rostock, D-18051 Rostock, Germany (Dated: January 15, 2020)The efficient certification of nonclassical effects of light forms the basis for applications in opticalquantum technologies. We derive general correlation conditions for the verification of nonclassicallight based on multiplexed detection. The obtained nonclassicality criteria are valid for imperfectlybalanced multiplexing scenarios with on-off detectors and do not require any knowledge about thedetector system. In this sense, they are fully independent of the detector system. In our experiment,we study light emitted by clusters of single-photon emitters, whose photon number may exceed thenumber of detection channels. Even under such conditions, our criteria certify nonclassicality withhigh statistical significance.
I. INTRODUCTION
The verification of quantum correlations in optical sys-tems is a key task in quantum optics. Besides its fun-damental significance for the understanding of radiationfields, the identification of genuine quantum features isbecoming ever more important as they can be used for ap-plications in quantum technologies [1–3]. A major goal,in this context, is to develop robust methods which ide-ally do not rely on any knowledge or assumptions aboutthe studied system, leading to the concept of device-independent quantum characterization [4–8].An important task is the characterization of light inthe few-photon regime. For the analysis of quantum lightin this regime, so-called multiplexing strategies [9–18]have been developed as a way of gaining insights in themeasured quantum state even when a photon-number-resolving measurement is not accessible. Such strategiesdo not provide a direct access to the photon-number dis-tribution and consequently the interpretation of the mea-surement statistics as the photon-number statistics canlead to a false certification of nonclassicality [19]. How-ever, nonclassicality criteria which can be directly ap-plied to the recorded click-counting statistics have beenformulated [20–25]. In particular, such criteria are veryefficient and successful in certifying nonclassicality fromexperimental data [26–33].One common assumption for such conditions is thatthe incoming light is equally split and detected in eachdetection channel. Recently, the detector-independentverification of quantum light for such equal splitting hasbeen reported [8, 34]. In some cases, however, an equalsplitting ratio might be hard to realize and requires thecareful characterization of the optical elements. Further-more, other multiplexing strategies such as fiber-loop de-tectors [11, 33] by design do not provide an equal split- ∗ [email protected] † [email protected] ting. For such unequal-splitting scenarios, a conditionbased on second-order moments [25] has been derivedas a generalization of the corresponding equal-splittingcondition [20]. More general higher order criteria have,however, not yet been reported for such unequal-splittingdetections.In this paper, we introduce detector-independent gen-eral (higher order) conditions for the certification of non-classical light measured with unbalanced multiplexingschemes and on-off detectors. The presented conditionsare fully independent of the properties of the used de-tection scheme. Based on Chebyshev’s integral inequal-ity, we derive a family of inequality conditions for theno-click events at the output channels which have to befulfilled for any classical radiation fields; their violationverifies nonclassicality. The so-obtained inequalities in-clude simple covariance conditions between two detec-tion channels and more general higher order correlationconditions. Our approach is based on minimal assump-tions and requirements which guarantees the applicabil-ity to any multiplexing setup, even without the knowl-edge about the used detectors and the splitting ratios.We demonstrate the strength of the obtained criteria bycertifying nonclassicality of light from clusters of single-photon emitters with high statistical significance. The re-lations of the presented nonclassicality certifiers to othernonclassicality criteria based on the Mandel Q parameterand the matrix of moments approach are discussed. II. MULTIPLEXING DETECTION
We are interested in the certification of quantum cor-relations based on general multiplexing scenarios, asschematically sketched in Fig. 1. The incident quantumstate of light is (unequally) split into N output channels.Each of these channels is then measured by a single on-off detector, which is the standard working principle ofmultiplexing detectors [9–18].Let us describe this multiplexing step formally. Weexpress the input quantum state in terms of the Glauber a r X i v : . [ qu a n t - ph ] J a n ... channel 1channel 2channel N U ... FIG. 1. Working principle of a multiplexing device. Theinput quantum state ˆ ρ is split at an unbalanced multiportsplitter and each output channel is measured with an on-offdetector. Sudarshan P representation [35, 36],ˆ ρ in = (cid:90) d αP ( α ) | α (cid:105)(cid:104) α | , (1)where | α (cid:105) is a coherent state. A quantum state iscalled classical if and only if (iff) its P function is non-negative [37, 38]. Multiplexing devices act as N × N multimode splitter and can be described via the uni-tary operation U ( N ) = ( u i,j ) Ni,j =1 which relates theinput to the output field operators via ˆ a out = U ( N )ˆ a in and ˆ a in(out) =(ˆ a , . . . , ˆ a N in(out) ). In the multiplex-ing case, only the first mode is occupied and the otherones are in the vacuum state. Consequently, the outputquantum state can be written asˆ ρ out = (cid:90) d αP ( α ) | u , α, . . . , u ,N α (cid:105)(cid:104) u , α, . . . , u ,N α | , (2)with (cid:80) k | u ,k | =1 [39]. We explicitly do not restrict ourconsideration to the case of uniform splitting, i.e., ∀ k : | u ,k | = 1 /N .We are now interested in the detector-click probabilityin each channel. The probability of detecting no click inthe k -th channel is given by the expectation value (cid:104) : ˆ m k : (cid:105) = (cid:90) d αP ( α ) (cid:104) u ,k α | : ˆ m k : | u ,k α (cid:105) , (3)where : . . . : denotes the normal-ordering prescription;cf. [40]. The corresponding operator is defined asˆ m k = e − ˆΓ k (ˆ n k ) , the subscript k indicates the detectionchannel and ˆ n k is the photon-number operator in thecorresponding channel. The probability of obtaining aclick in this channel is given by 1 −(cid:104) : ˆ m k : (cid:105) . The detectorresponse function ˆΓ k (ˆ n k ) is a function of ˆ n k and describesthe connection between the electromagnetic field and thegeneration of a click [40, 41]. The detector response func-tion can be determined via direct calibration techniques;see, e.g., [42]. III. CONDITIONS FOR QUANTUMCORRELATIONS
We aim at formulating nonclassicality conditions basedon the correlations between the no-click events of thedifferent output channels which do not depend on the characteristics of the used multiplexing architecture anddetectors. The simplest case we can consider in this con-text is the correlation between the no-click events of twooutput channels [43]. In fact, we can use Chebyshev’sintegral inequality (see, e.g., [44]) to derive the simplecondition, (cid:104) :Cov( ˆ m i , ˆ m j ): (cid:105) = (cid:104) : ˆ m i ˆ m j : (cid:105) − (cid:104) : ˆ m i : (cid:105)(cid:104) : ˆ m j : (cid:105) cl ≥ , (4)which must hold for any classical input state, i.e., a quan-tum state with a non-negative P function. Details on thederivation are provided in the Appendix A. The violationof this inequality is a direct and experimentally easily ac-cessible signature of nonclassicality and is directly relatedto the negativities of the P function of the studied state.In particular, only one multiplexing step and on-off de-tection is sufficient for the application of condition (4).Condition (4) has a clear physical interpretation. Non-negative normal-ordered covariances can be explained interms of a classical description of the measured radiationfield, i.e., by a classical P function. In particular, for aninput coherent state, the no-click events are uncorrelated[ (cid:104) :Cov( ˆ m i , ˆ m j ): (cid:105) =0], which represents the boundary be-tween classical and nonclassical radiation fields. On theother hand, an anticorrelation, i.e., a negative covarianceof the no-click events, can only arise from negativities inthe P function of the considered state. For example, asingle-photon input state leads to an anticorrelation ofthe no-click events which is revealed by a negative co-variance.We can further generalize condition (4) to multimodehigher order moment conditions. Again by making useof Chebyshev’s integral inequality [44], we formulate thefamily of correlation conditions (see Appendix A for de-tails) (cid:104) : ˆ m I . . . ˆ m I K : (cid:105) − (cid:104) : ˆ m I : (cid:105) · · · (cid:104) : ˆ m I K : (cid:105) cl ≥ , (5)where I . . . I K are mutually disjoint subsets (partitions)of I = { , . . . , N } and ˆ m J is the no-click operator for alldetection channels in I J , ˆ m I J = (cid:81) j ∈I J ˆ m j . These gen-eral conditions also include asymmetric partitions andthe clustering of channels. Note that Eq. (5) generalizesthe approach in [23] to unequal splitting, uncharacterizeddetectors, and arbitrary partitions. In Appendix A, weshow that the conditions (4) and (5) are not affected bydark counts or other uncorrelated noise. IV. EXAMPLE
Before turning to the experiment and the dataanalysis, let us consider an example. As an in-put state, we choose an n -photon-added thermal state N n (cid:0) ˆ a † (cid:1) n ˆ ρ th ˆ a n , where the thermal state is ˆ ρ th = 1 / ( n +1) (cid:80) ∞ k =0 ( n/ ( n + 1)) k | k (cid:105)(cid:104) k | with N n being the normal-ization constant [45]. Such states have been realized inexperiments [46, 47]. We consider a single unbalanced nonclassicality FIG. 2. The no-click-covariance condition (4) (solid lines;scaled by a factor of 5) and the photon-number covariance[Eq. (6)] (dashed) are shown for one- and two-photon-addedthermal states in dependence on the mean thermal photonnumber n . multiplexing step (beam splitter) with an intensity split-ting of 70:30 and a detection efficiency of 0 .
7. By vio-lating the covariance inequality (4), we can certify non-classicality. We compare this nonideal detection schemewith a classicality condition based on the photon-numbercovariance, (cid:104) ˆ n i ˆ n j (cid:105) − (cid:104) ˆ n i (cid:105)(cid:104) ˆ n j (cid:105) cl ≥ , (6)where ˆ n j is the photon-number operator in the j th de-tection channel. The application of this condition wouldrequire experimental access to the second-order momentsof the photon-number operator. The violation of thiscondition corresponds to the nonclassicality condition interms of the second-order intensity correlation function, g (2) (0) <
1, which is closely related to the sub-Poissonianphoton statistics and the Mandel Q parameter [48].In Fig. 2, the behavior of the two conditions is shownfor one and two-photon-added thermal states in depen-dence on the thermal photon number. The violationof the no-click inequality (4) detects nonclassicality ina wider range of n than corresponding violation of thephoton-number condition (6). Similar behaviors havebeen observed for the sampling of phase-space distribu-tions from multiplexing detection data [24, 31].An explanation for this behavior can be found whenone considers the measurement operator of the click de-tection, i.e., the no-click operator ˆ m k . This operator isan exponential function of the photon-number operatorand, thus, higher order moments of the photon-numberoperator contribute to the condition (4). Therefore, theclick detection may be more sensitive toward nonclassicaleffects than the detection of the first two moments of thephoton-number operator in Eq. (6). V. EXPERIMENT
In our experimental setup, we study light from clustersof single-photon emitters which we detect with the helpof time-bin multiplexed click detection. We used multi-photon light emitted by clusters of colloidal CdSe/CdSquantum ”dot in rods” (DRs) [49–52] coated on a fused
D1D2O1 DR O2 F
C1 C2C3 C4
FIG. 3. Principal scheme of the setup. The pump radiation(355 nm) is focused into a cluster of DRs through objectivelens O1 and then cut off by the filters F. The radiation emit-ted by the cluster is coupled into an objective lens O2 and,after filtering, is sent into the fiber-assisted multiplexed de-tection setup where two detectors D1, D2 and two differentpath lengths create four detection channels, C1, . . . , C4. silica substrate and excited by picosecond pulses at 355nm. For a detailed description of the experiment, seeRef. [53]. To get rid of the pump radiation, the emittedlight was filtered using a long-pass filter and a band-pass filter (center wavelength 607 nm, bandwidth 42nm). The size of the cluster was determined by as-suming that the mean number of photons emitted perpulse scales with the number of emitters in the cluster.In this way, we obtained clusters with an effective sizebetween 2 and 14 emitters. Each DR in such a clus-ter, provided that it is excited, emits a quantum statethat is close to a single-photon one, with an extremelysmall admixture of a two-photon component. Takinginto account the 25% excitation probability per excita-tion pulse, the state emitted by a single DR can be writ-ten as ˆ ρ DR = p | (cid:105)(cid:104) | + p | (cid:105)(cid:104) | + p | (cid:105)(cid:104) | , where p ≈ . p ≈ .
1, and p ≈ − . The low probability of two-photonemission leads to strongly nonclassical g (2) (0) ≤ .
05. Al-though different DRs in a cluster emit incoherently, theresulting state manifests nonclassicality because the totalnumber of emitted photons is restricted according to thesize of the cluster [53, 54].The radiation emitted by a single cluster was collectedwith an efficiency of 44%, which takes into account thelosses at all optical elements, and sent to a fiber multi-plexed detection setup, cf. Fig. 3, comprising two clickdetectors based on avalanche photodiodes, with the quan-tum efficiency 60%. The use of fiber loops provided twotime bins for each detector, and therefore the setup wasequivalent to the one shown in Fig. 1, with N =4 chan-nels. For each of the studied clusters, we collected adataset containing between 10 and 10 pulses, and foreach pulse, the number of click counts in each channelwas registered. Depending on the size of the cluster, themean click number per pulse was between 0.05 and 0.1due to the low excitation and detection efficiency. VI. RESULTS
We apply the nonclassicality criteria based on the in-equalities (4) and (5) to the multiplexing data obtainedfor the different cluster sizes. Our approach can be di-rectly applied to the measured data without the knowl- cluster size a)b) 2 4 6 8 10 12 140- 1- 2- 3 nonclassicality nonclassicalitycluster sizecluster size a)b) 2 4 6 8 10 12 140- 1- 2- 3 nonclassicality nonclassicalitycluster size
FIG. 4. a) The covariance condition (4) between two de-tection channels (upper, orange markers) and the conditionbased on the full partition of all channels, cf. Eq. (5) (lower,green markers), scaled by a factor of 10 , are shown togetherwith their sampling errors (gray bars) for different clustersizes. The dashed and dotted lines represent the correspond-ing model, Eqs. (7) and (8) with η = 0 .
009 and different M (the lines should guide the eye). b) The behavior for highernumbers of emitters following the theoretical model. The non-classicality conditions only approach the classical limit forhigh number of emitters due to saturation of all detectors. edge of the detection system or data post-processing. Weanalyze the correlation condition (4) between two of thedetection channels (first and third) and the conditioncorresponding to the full partition of all four detectionchannels [cf. Eq. (5)] for the different cluster sizes. Theno-click moments and their statistical errors can be di-rectly sampled from the measurement statistics [27].The results are shown in Fig. 4 a). Even the sim-ple covariance condition (4) between two of the outputchannels is capable of certifying nonclassicality with sta-tistical significances (estimated value divided by its sta-tistical sampling error) above three standard derivationsfor all cluster sizes. Furthermore, we identify that thecondition based on the full partition of the no-click op-erator into four parts [cf. Eq. (5)] even yields verifica-tions with higher statistical significances of up to 29 . η . The lightemitted by a cluster of M incoherently emitting quantumdots is an M -mode tensor-product state of single-photonstates. Furthermore, we assume for simplicity symmet-ric multiplexing, and that the M -mode state is mode-insensitively recorded by the detectors. This yield theexpressions of the conditions (4) and (5), (cid:104) :Cov( ˆ m , ˆ m ): (cid:105) = (cid:16) − η (cid:17) M − (cid:16) − η (cid:17) M and (7) (cid:104) : ˆ m . . . ˆ m : (cid:105) − (cid:89) i =1 (cid:104) : ˆ m i : (cid:105) = (1 − η ) M − (cid:16) − η (cid:17) M (8)respectively; cf. Appendix B for more details. This sim-ple model captures the dependence on the cluster size[Fig. 4 a)] and explains that for the relatively low ef-ficiency of the experiment ( η ≈ . η determineshow fast the classical limit is reached. Examples withother efficiencies are provided in Appendix B. Further-more, we can conclude that with the proposed criteria itwould be possible to certify nonclassicality of the lightemitted by clusters of several hundreds of emitters withthe used experimental setup. VII. DISCUSSION
The obtained covariance condition (4) is closelyconnected to the Mandel Q parameter [48] and re-lated second-order moment conditions such as the sub-binomial Q B [20] and sub-Poisson-binomial Q PB [25] pa-rameters. In fact, we show in Appendix C that these cri-teria can be traced back to condition (4). Thus, we couldidentify the covariance condition (4) as the fundamentalbuilding block of these other nonclassicality criteria.Let us now consider conditions including higher or-der moments, such as in Eq. (5). This includes alsoasymmetric partitions of the corresponding no-click op-erators, i.e., partitions in which different parts may havedifferent number of elements, such as, e.g., the parti-tion (cid:104) : ˆ m ˆ m : (cid:105)(cid:104) : ˆ m : (cid:105) . Other approaches [8, 21, 27, 28]are based on the matrix of moments which, by construc-tion, cannot involve asymmetric-partition conditions. InAppendix C, a comparison of these methods with thederived conditions is presented. Hence, the here derivedconditions are by construction different from already ex-isting approaches and provide a wider applicability [55].A detector-independent method for the certification ofquantum light through multiplexing was already intro-duced [8]. In that work, the detector independence referssolely to the detectors and not to the whole detectionsystem, including the multiplexing, as an equal splittinginto the detection channels is required. With our ap-proach, we can relax the latter requirement and, hence,have a condition which is fully independent of the wholedetection scheme.Furthermore, we would like to discuss the relationof the introduced nonclassicality conditions to entangle-ment. The derived conditions can certify nonclassical-ity of the input quantum state of a multiplexing device.In Sec. IV of Ref. [56] it was undoubtedly demon-strated that the multiplexing of a nonclassical state oflight yields multipartite entangled states. This is justwhat happens in our experiment. However, the directcertification of the entanglement requires an extensionof the measurement setup, which is beyond the scope ofthis paper. Although we are testing nonclassicality of theinput state, our conditions in fact reveal quantum corre-lations between the different detection channels. Hence,the obtained nonclassicality conditions are closely relatedto multimode entanglement. This opens possibilities forfuture applications in quantum technologies. VIII. CONCLUSIONS
We formulated conditions for the verification of non-classical light detected by general multiplexing setups. The obtained criteria do not require any knowledge aboutsplitting ratios and used detectors. Hence, our meth-ods are fully independent of the detection scheme. Theobtained correlation criteria include conditions based onsecond-order and higher order moments of the no-clickevents. We demonstrated the strength of our approachby certifying nonclassicality of light emitted by clustersof single-photon emitters. Importantly, the presented cri-teria are capable of detecting quantum light even if thenumber of photon emitters is higher than the number ofused detection channels.We could show that our conditions based on on-offdetectors can provide even more insight into the non-classical character of the recorded light than compara-ble approaches based on ideal photon-number-resolvingmeasurements. Furthermore, we discussed the relationto established nonclassicality indicators and showed thatour approach provides new forms of nonclassicality con-ditions which cannot be deduced from other existingcriteria. The present results provide useful and simpletools for the detector-independent verification of quan-tum light, applicable to many experimental scenarios.
ACKNOWLEDGMENTS
This work has been supported by DeutscheForschungsgemeinschaft through Grant No. VO501/22-2. MB acknowledges financial support by theLeopoldina Fellowship Programme of the German Na-tional Academy of Science (LPDS 2019-01) and thanksValentin Gebhart and Elizabeth Agudelo for helpfulcomments.
Appendix A: Derivation of the nonclassicality conditions
Here we will show how the introduced nonclassicality conditions can be derived by using Chebyshev’s integralinequality. Furthermore, we demonstrate that the results of the obtained conditions are not influenced by uncorrelatednoise sources.
1. Chebyshev’s integral inequality
In order to construct the nonclassicality conditions we will make use of Chebyshev’s integral inequality; see, e.g., [44].Let f and g be two functions which are integrable and monotone in the same sense on ( a, b ) and let p be a positiveand integrable function on the same interval. Then the Chebyshev’s integral inequality (cid:90) ba p ( x ) f ( x ) g ( x ) dx (cid:90) ba p ( x ) dx ≥ (cid:90) ba p ( x ) f ( x ) dx (cid:90) ba p ( x ) g ( x ) dx, (A1)holds. In the following, we will see that we can make use of this inequality for the derivation of the nonclassicalityconditions. In our case, the p ( x ) will be the (phase-averaged) P function of a classical quantum state and f , g willbe the expectation values of the normal-ordered no-click operators with coherent states.
2. Two-channel no-click correlation
The starting point of this consideration is the multimode state after the multiplexing step,ˆ ρ out = (cid:90) d αP ( α ) | u , α, . . . , u ,N α (cid:105)(cid:104) u , α, . . . , u ,N α | , (A2)with (cid:80) k | u ,k | = 1. We consider the no-click operators ˆ m i = e − Γ(ˆ n i ) whose normally ordered expectation valueswith coherent states, (cid:104) β | : ˆ m i | β : (cid:105) = (cid:104) β | : e − Γ(ˆ n i ) : | β (cid:105) = e − Γ( | β | ) , are monotonically decreasing functions of | β | . A typicalexample of a detector response function is a linear response function Γ = η i | β | + ν i , where η i and ν i are the quantumefficiency and the dark-count rate in the i th mode, respectively.Then, the normal-ordered expectation value of the product of the two no-click operators ˆ m i and ˆ m j reads as (cid:104) : ˆ m i ˆ m j : (cid:105) = Tr[ˆ ρ out : ˆ m i ˆ m j :] = (cid:90) d αP ( α ) e − Γ i ( | u ,i | | α | ) e − Γ j ( | u ,j | | α | ) . (A3)As both functions e − Γ( | u ,i | | α | ) and e − Γ( | u ,i | | α | ) are monotonically decreasing functions of | α | and we assume anon-negative (classical) P distribution, we can apply Chebyshev’s integral inequality and obtain (cid:90) d αP ( α ) e − Γ i ( | u i | | α | ) e − Γ j ( | u j | | α | ) ≥ (cid:90) d αP ( α ) e − Γ i ( | u i | | α | ) (cid:90) d αP ( α ) e − Γ j ( | u j | | α | ) , (A4)which can be written in terms of the covariance, (cid:104) :Cov( ˆ m i , ˆ m j ): (cid:105) = (cid:104) : ˆ m i ˆ m j : (cid:105) − (cid:104) : ˆ m i : (cid:105)(cid:104) : ˆ m j : (cid:105) cl ≥ . (A5)This inequality has to hold for any non-negative (classical) P and, thus, a violation of the inequality immediatelyuncovers nonclassicality of the considered state. Let us stress that the derivation does not rely on the knowledge ofthe splitting ratios and the properties of the detectors.
3. Multimode generalization
Here, we will show how the multimode generalizations of the two-mode covariance condition can be obtained byapplying Chebyshev’s integral inequality several times. To derive the multimode conditions for N detection modes,we make several times use of Chebyshev’s integral inequality. We start from the expectation value (cid:104) : N (cid:89) i =1 ˆ m i : (cid:105) = (cid:90) d αP ( α ) N (cid:89) i =1 e − Γ i ( | u ,i | | α | ) , (A6)which can be written as (cid:104) : N (cid:89) i =1 ˆ m i : (cid:105) = (cid:90) d αP ( α ) e − (cid:80) Ni =1 Γ i ( | u ,i | | α | ) = (cid:90) d αP ( α ) e − (cid:80) i ∈I Γ i ( | u ,i | | α | ) e − (cid:80) i ∈I Γ i ( | u ,i | | α | ) (A7)where I and I are two bipartitions of the considered operator functions. Note that both e − (cid:80) i ∈I Γ i ( | u ,i | | α | ) and e − (cid:80) i ∈I Γ i ( | u ,i | | α | ) are monotonically decreasing functions. As in the case above, we can now apply Chebyshev’sintegral inequality which yields (cid:90) d αP ( α ) e − (cid:80) Ni =1 Γ i ( | u ,i | | α | ) ≥ (cid:90) d αP ( α ) e − (cid:80) i ∈I Γ i ( | u ,i | | α | ) (cid:90) d αP ( α ) e − (cid:80) i ∈I Γ i ( | u ,i | | α | ) , (A8)which holds for any non-negative P function. This may also be written as (cid:104) : (cid:81) Ni =1 ˆ m i : (cid:105) ≥ (cid:104) : (cid:81) i ∈J ˆ m i : (cid:105)(cid:104) : (cid:81) j ∈J ˆ m j : (cid:105) .This procedure can be repeated several times which leads to the general form (cid:104) : ˆ m I . . . ˆ m I K : (cid:105) − (cid:104) : ˆ m I : (cid:105) · · · (cid:104) : ˆ m I K : (cid:105) cl ≥ , (A9)where I . . . I K are mutually disjoint subsets (partitions) of I = { , . . . , N } and ˆ m J is the no-click operator for alldetection channels in I J , ˆ m I J = (cid:81) j ∈I J ˆ m j . A full partition of the non-click operators yields the condition (cid:104) : ˆ m . . . ˆ m N : (cid:105) − (cid:104) : ˆ m : (cid:105) · · · (cid:104) : ˆ m N : (cid:105) cl ≥ . (A10)
4. Independence of uncorrelated noise contributions
Furthermore, we can show that the derived covariance conditions do not depend on uncorrelated noise contributionssuch as detector dark counts. Let us assume that two detectors have linear detector responses which can be describedby the functions ˆΓ i ( j ) (ˆ n i ( j ) ) = η i ( j ) ˆ n i ( j ) + ν i ( j ) , where η i ( j ) and ν i ( j ) are the quantum efficiencies and the uncorrelatednoise-count rates, respectively. In this case, the normal-ordered expectation values of the no-click operators are (cid:104) : ˆ m i ˆ m j : (cid:105) = e − ( ν i + ν j ) (cid:82) d αP ( α ) e − η i | u i | | α | e − η j | u j | | α | and (cid:104) : ˆ m i ( j ) : (cid:105) = e − ν i ( j ) (cid:82) d αP ( α ) e − η i ( j ) | u i ( j ) | | α | . As above,we can now derive an inequality which can be written in the form e − ( ν i + ν j ) (cid:20)(cid:90) d αP ( α ) e − η i | u i | | α | e − η j | u j | | α | − (cid:90) d αP ( α ) e − η i | u i | | α | (cid:90) d αP ( α ) e − η j | u j | | α | (cid:21) cl ≥ e − ( ν i + ν j ) (cid:104) :Cov( ˆ m i , ˆ m j ): (cid:105) FUN cl ≥ , (A12)where (cid:104) :Cov( ˆ m i , ˆ m j ): (cid:105) FUN corresponds to the covariance in the case when the detected light field would be free ofany uncorrelated noise (FUN). We see that the uncorrelated noise contributions only result in an overall scaling ofthe inequality but do not influence the sign of the inequality. Therefore, the uncorrelated noise contributions do notalter the corresponding nonclassicality verification. This consideration can be straightforwardly generalized to themultimode conditions. Note that a similar independence of dark-count contributions has been reported in the contextof phase-sensitive measurements with multiplexing detectors [57].
Appendix B: Radiation state and detection model for clusters of single-photon emitters
We derive the model for the multiplexed detection of light from a cluster of incoherently emitting single-photonemitters. This model is used to explain and interpret the experimental results for the different cluster sizes presentedin Fig. 4 of the main text. We consider clusters consisting of M quantum-dot emitters and we assume that allquantum dots have equal properties and that each quantum dot emits a single photon. Thus, the light emitted bythe cluster can be expressed by a tensor-product state of single-photon states, i.e., | χ (cid:105) = (cid:81) Mi =1 ⊗| (cid:105) i where the upperindex indicates the different single-photon modes. Note that the no-click operators of the different detection channelsare labeled with a lower index. In the experiment each quantum dot is excited with a certain finite probability, so thatunexcited quantum dots do not contribute to the single photon emission. We note that a quantum dot which doesnot emit a photon is equivalent to one emitting a photon but the photon gets lost, i.e., it is not recorded. Therefore,we consider that each quantum dot emits a single photon but we assign a quantum efficiency η ex in the detection ofthe light which accounts for the finite probability of exciting each quantum dot. Furthermore, not all light emittedby the clusters is collected which can be modeled by the collection efficiency η col . We assume that the multiplexeddetection is symmetric and that the on-off detectors have a linear detector response which is characterized throughthe detector efficiency η det . All the different efficiencies can be summarized in the total efficiency of the experiment, η = η ex η coll η det . Furthermore, the detection in each channel is not mode sensitive as all impinging photons can lead toa detection click. Therefore, we can write the no-click operator of the j -th detection channel as ˆ m j = exp[ η ( (cid:80) Mi =1 ˆ n ij )].Let us now consider the multiplexing of the tensor product single-photon states. The multiplexing device actsequally on each single-photon state and we assume a symmetric splitting into the four detection channels which yields M (cid:89) i =1 ⊗| (cid:105) i → M (cid:89) i =1 ⊗ (cid:20) √ (cid:0) | , , , (cid:105) i + | , , , (cid:105) i + | , , , (cid:105) i + | , , , (cid:105) i (cid:1)(cid:21) . (B1)Now we calculate the expectation value of the no-click operator in the j -th detection channel to be (cid:104) ˆ m j (cid:105) = (1 − η ) M .Similarly, the expectation value of the joint no-click event for all four detection channels is given by (cid:104) : ˆ m . . . ˆ m : (cid:105) =(1 − η ) M . This allows us to evaluate the expressions of the condition (4) and (A10) ( N = 4) to be (cid:104) : ˆ m ˆ m : (cid:105) − (cid:104) : ˆ m : (cid:105)(cid:104) : ˆ m : (cid:105) = (1 − η M − [(1 − η M ] (B2) (cid:104) : ˆ m . . . ˆ m : (cid:105) − (cid:89) i =1 (cid:104) : ˆ m i : (cid:105) = (1 − η ) M − [(1 − η M ] , (B3) FIG. 5. The values of the two nonclassicality conditions scaled by a factor of 10 for the state and detection model, Eq. (7)(dotted) and Eq. (8) (dashed), are shown in dependence on the number of emitters for η = 0 .
05 (a) and η = 0 . respectively, which are given as as Eqs. (7) and (8) in the main text. These results are used in Fig. 4 of the mainpaper to explain the dependence of the nonclassicality conditions on the cluster size.Let us note that the only parameter in this model is the overall efficiency η which accounts for all inefficiencies ofthe whole setup. For the analysis of our experimental data we estimate this efficiency to be η ≈ . η is exemplarily shown in Fig. 5. We observe that they all follow the same behavior, i.e., that the nonclassicalitycondition is at first getting more negative with increasing numbers of emitters, reaches its minimum for a certainnumber of emitters M min( η ) , and eventually approaches the classical limit of zero for large numbers of emitters. Theposition of its minimal value is determined by the overall efficiency η . In the limit of unity efficiency ( η → M = 1) which is the manifestation of the single-photon character and theresulting anticorrelations in the click events (”antibunching”). However, we see that for lower efficiencies η we canstill certify nonclassicality even for large numbers of single-photon emitters. Remarkably, this is possible even if thenumber of detection channels ( N = 4) is orders of magnitude smaller than the numbers of emitters; cf. Fig 5 and Fig.4 in the main text. We note that the presented model is rather simple and that it is based on several assumptions.This explains deviations of the model from some of the experimental data points while it describes well the overalldependence on the cluster size. Appendix C: Relation to other nonclassicality conditions
In this section, we show some relations and differences to established methods for the verification of nonclassicalityfrom multiplexing measurements. First, we will discuss the relation to conditions of the form of the Mandel Q parameter. Second, we show that our approach delivers criteria which cannot be obtained via the matrix of momentsapproach.
1. Relation to conditions of the form of the Mandel Q parameter Here, we will show how the obtained covariance condition (4) relates to established nonclassicality parameters ofthe form of the Mandel Q parameter [48]. Besides the Mandel parameter itself, we will also consider the recentlyintroduced sub-binomial Q B [20] and sub-Poisson-binomial Q PB [25] parameters. The latter two were introduced forthe verification of nonclassical light in symmetric and general (asymmetric) multiplexing schemes, respectively. Allthree parameter have in common that—whenever they attain negative values—they uncover nonclassicality of theconsidered quantum state. a. Sub-Poisson-Binomial parameter Let us start by considering the sub-Poisson-Binomial parameter. The sub-Poisson-binomial parameter [25] may bewritten as Q PB = (cid:80) Ni (cid:54) = j (cid:104) :Cov( ˆ m i ˆ m j ): (cid:105) (cid:80) Ni (cid:104) : ˆ m i : (cid:105) (1 − (cid:104) : ˆ m i : (cid:105) ) , (C1)where N is the number of detection channels and (cid:104) :Cov( ˆ m i ˆ m j ): (cid:105) is the normal-ordered expectation value of thecovariance defined in Eq. (4). Q PB < Q PB characterizes nonclassicality if the sum over all possible (cid:104) :Cov( ˆ m i ˆ m j ): (cid:105) is negative and, thus, it is based on the violation of the simple classical covariance condition (4).However, it is a more complex condition as it requires evaluation of all possible correlations between the differentdetection channels. b. Sub-Binomial parameter If we now assume that the multiplexing is performed with an equal splitting into the channels and each channel isdetected with detectors which have equal properties, i.e., having the same detector response (quantum efficiency anddark-count rate), the Q B parameter [20], Q B = ( N − (cid:104) :Var( ˆ m ): (cid:105)(cid:104) : ˆ m : (cid:105) (1 − (cid:104) : ˆ m : (cid:105) ) , (C2)may be applied. A negative value of Q B uncovers nonclassicality. Note that Q B is the special form of Q PB in the caseof equal splitting and detection. In this case, all no-click operators ˆ m i are equal and we can replace them by ˆ m . Then,the covariance in Eq. (C1) reduces to the variance in Eq. (C2). Still the negativities of Q B arise from a negativevariance, which is nothing else as the violation of the classical covariance condition (4) for the case in which all ˆ m i are the same. The negativity of the variance is also a direct indication that the corresponding quantum state cannotbe described by a classical (non-negative) P function. Note, however, that Q B is only applicable if the assumptionsof equal splitting and detection are fulfilled. c. Mandel parameter In [20], it has been shown that Q B approaches the Mandel Q parameter Q = (cid:104) :Var(ˆ n ): (cid:105)(cid:104) :ˆ n : (cid:105) , (C3)in the case of an infinite number of detection channels, i.e., lim N →∞ Q B = Q . In other words, for an equal splittinginto an infinite number of on-off detectors the variance of the no-click operator yields the normal-ordered expectationvalue of the variance of the photon-number operator, (cid:104) :Var(ˆ n ): (cid:105) . Hence, Q can be seen as the limiting case of recordinglight with an infinite number of equal on-off detectors. This finding agrees with a derivation of the photon-countingformula in the 1960s [58], where a bulk material described by an infinite number of single atoms (acting in the sameway as on-off detectors) detects the light.We can summarize that criteria of the form of the parameters Q , Q B , and Q PB are in the end based on thecovariance condition in Eq. (4). Importantly, the condition in Eq. (4) is the most simple form of such conditions andprovides the essential building block for the other criteria. Moreover, the simple covariance condition using only twoon-off detectors can be more sensitive than the condition provided by the Q parameter, as we show in Fig. (2), forthe example of a photon-added thermal state.
2. Relation to criteria based on the matrix of moments approach
Here, we compare the obtained conditions with the matrix of moments approach for multiplexing detection [21].For classical states, the matrix of moments M is positive semidefinite, with0 cl ≤ M = (cid:16) (cid:104) : ˆ m l + l (cid:48) : (cid:105) (cid:17) (cid:98) N/ (cid:99) l,l (cid:48) =0 . (C4)where the floor function yields (cid:98) N/ (cid:99) = N/ N and (cid:98) N/ (cid:99) = ( N − / N . A violation of thispositive semidefiniteness would imply that the measured quantum state is a nonclassical one. As an example we canconsider the 2 × l th order ( l ≥ (cid:18) (cid:104) : ˆ m : (cid:105) (cid:104) : ˆ m l : (cid:105)(cid:104) : ˆ m l : (cid:105) (cid:104) : ˆ m l : (cid:105) (cid:19) = (cid:104) :Var( ˆ m l ): (cid:105) cl ≥ . (C5)0This, in fact, corresponds to the correlation conditions derived here if all channels are recorded equally. It is, however,important to mention that, contrary to the conditions presented here, the matrix of moments approach can only beapplied if equal splitting and detection are considered.We have seen that for some cases (equal splitting and detection, and 2 × (cid:104) : ˆ m k : (cid:105) − (cid:104) : ˆ m : (cid:105)(cid:104) : ˆ m k − : (cid:105) cl ≥ k > (cid:104) : ˆ m k : (cid:105) − (cid:104) : ˆ m : (cid:105)(cid:104) : ˆ m k − : (cid:105)(cid:104) : ˆ m : (cid:105) cl ≥ k > . (C7)Hence, the approach presented here provides new conditions which are not covered by the matrix of moments approach.Therefore, these more general conditions might be able to certify nonclassicality in cases where other methods fail todo so. [1] S. L. Braunstein and P. van Loock, Quantum informa-tion with continuous variables, Rev. Mod. Phys. , 513(2005).[2] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P.Dowling, and G. J. Milburn Linear optical quantum com-puting with photonic qubits Rev. Mod. 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